So what's the benefit of using one over the other? The way I understand it, in Av or Tv, you only have to adjust the shutter or aperture, and the rest is done automatically for you...is this correct? Is there some advantage shooting with Av/Tv that I just don't know about?

The advantage of aperture or shutter priority modes over manual is the same as any automation: when it works right, it takes labor out of our hands and makes life easier, allowing us to concentrate on other things. And on modern cameras, the automatic metering is pretty good — it basically does the right thing most of the time. And, if you get to know your camera, you can judge pretty well when it won't.

The advantage of these modes over full automatic ("P" mode, usually; for "program") is that there is more than one way to arrive at any given final exposure value: you can mix and match aperture, shutter speed, and ISO in several different ways with the same final result for exposure, but with very different effects on the composition. (Briefly: faster shutter freezes motion; stopped-down aperture gives more depth of field; lower ISO reduces noise.) When you use full automatic, you get whatever (usually middle-of-the-road) balance of these things that the auto-exposure program works out. Av or Tv (or Sv, for ISO priority, on some cameras), let you choose — and directly control — the aspect you care about most for composition, while letting the camera work out the exposure to match the metering.

The Av & Tv modes are where the camera sets the settings it thinks is correct for the shot. The manual setting allows the photographer to have the control of all the settings to produce the shot wanted.